## Asymmetric Encryption-

Before you go through this article, make sure that you have gone through the previous article on Asymmetric Key Cryptography.

In asymmetric encryption,

• Sender and receiver use different keys to encrypt and decrypt the message.
• The famous asymmetric encryption algorithms are-

## Symmetric Key Cryptography-

In symmetric key cryptography,

• Both sender and receiver use a common secret key to encrypt and decrypt the message.
• The major issue is exchanging the secret key between the sender and the receiver.
• Attackers might intrude and know the secret key while exchanging it.

## Diffie Hellman Key Exchange-

As the name suggests,

• This algorithm is used to exchange the secret key between the sender and the receiver.
• This algorithm facilitates the exchange of secret key without actually transmitting it.

## Diffie Hellman Key Exchange Algorithm-

Let-

• Private key of the sender = Xs
• Public key of the sender = Ys
• Private key of the receiver = Xr
• Public key of the receiver = Yr

Using Diffie Hellman Algorithm, the key is exchanged in the following steps-

### Step-01:

• One of the parties choose two numbers ‘a’ and ‘n’ and exchange with the other party.
• ‘a’ is the primitive root of prime number ‘n’.
• After this exchange, both the parties know the value of ‘a’ and ‘n’.

### Step-02:

• Both the parties already know their own private key.
• Both the parties calculate the value of their public key and exchange with each other.

 Sender calculate its public key as-Ys = aXs mod nReceiver calculate its public key as-Yr = aXr mod n

### Step-03:

• Both the parties receive public key of each other.
• Now, both the parties calculate the value of secret key.

 Sender calculates secret key as-Secret key = (Yr)Xs mod nReceiver calculates secret key as-Secret key = (Ys)Xr mod n

Finally, both the parties obtain the same value of secret key.

## Problem-01:

Suppose that two parties A and B wish to set up a common secret key (D-H key) between themselves using the Diffie Hellman key exchange technique. They agree on 7 as the modulus and 3 as the primitive root. Party A chooses 2 and party B chooses 5 as their respective secrets. Their D-H key is-

1. 3
2. 4
3. 5
4. 6

## Solution-

Given-

• n = 7
• a = 3
• Private key of A = 2
• Private key of B = 5

### Step-01:

Both the parties calculate the value of their public key and exchange with each other.

Public key of A

= 3private key of A mod 7

= 32 mod 7

= 2

Public key of B

= 3private key of B mod 7

= 35 mod 7

= 5

### Step-02:

Both the parties calculate the value of secret key at their respective side.

Secret key obtained by A

= 5private key of A mod 7

= 52 mod 7

= 4

Secret key obtained by B

= 2private key of B mod 7

= 25 mod 7

= 4

Finally, both the parties obtain the same value of secret key.

The value of common secret key = 4.

Thus, Option (B) is correct.

## Problem-02:

In a Diffie-Hellman Key Exchange, Alice and Bob have chosen prime value q = 17 and primitive root = 5. If Alice’s secret key is 4 and Bob’s secret key is 6, what is the secret key they exchanged?

1. 16
2. 17
3. 18
4. 19

## Solution-

Given-

• n = 17
• a = 5
• Private key of Alice = 4
• Private key of Bob = 6

### Step-01:

Both Alice and Bob calculate the value of their public key and exchange with each other.

Public key of Alice

= 5private key of Alice mod 17

= 54 mod 17

= 13

Public key of Bob

= 5private key of Bob mod 17

= 56 mod 17

= 2

### Step-02:

Both the parties calculate the value of secret key at their respective side.

Secret key obtained by Alice

= 2private key of Alice mod 7

= 24 mod 17

= 16

Secret key obtained by Bob

= 13private key of Bob mod 7

= 136 mod 17

= 16

Finally, both the parties obtain the same value of secret key.

The value of common secret key = 16.

Thus, Option (A) is correct.

To gain better understanding about Diffie Hellman Key Exchange Algorithm,

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